First off, hats off to you for rescuing a horse!
Second, congrats on getting such a rare little girl.. Especially since she is pony sized!!
Now back to your original question....
I am assuming you are not intending on going pro with this horse, she is a bit older than what is typical in the big times... But I can say which should improve with weight and exercise her rump is very little. QH have big ol' rumps which gives them power to come out of the barrels. I would really work on getting some groceries on that girl and maybe supplement with a flax seed oil to help build up with out creating fat deposits. Even at a trot she is being asked to exercise beyond her bodies physical allowance. She may do it becuase she has heart but it is still not good for her at this time. So maybe just a walk through the pattern until she gains some weight. A walk through the pattern is just as good as a trot but less physically demanding. When I train my barrel horses I do not run them at all for the first year. They are walked through the pattern 5 or 6 times ( I have some tricks on how to get them to setup at a walk) then go work on something else. I do start my horses younger at about 4 years old but they never run or trot a pattern for a year.
I can't see what kind of flexion she has which would be another important thing for her to have. Ground work her on her flexion to improve it ( I do it all the time with my horses even the ones who have great flexion) and if possible ride her up steep grades and make her walk. It will help build up those hind muscles. Stepping over things will also get those muscles in good shape (like logs and things where she really has to pick her feet up).
When you look at her rump it is not round it is almost triangle in shape which shows a serious lack of muscle there which she will need in order to power out of the turns. Get that bum looking better and overall she will be ok.